Phase-Locked Loops for Wireless Communications: Digital, Analog and Optical Implementations, Second Edition

2.6: Second Order Phase-Locked Loops

2.6 Second Order Phase-Locked Loops

The first order loop analysis for the three different inputs suggests a general equation for the loop filter,


The variable n, represents the desired order of the phase-locked loop. Jaffe and Rechtin [17] investigated the optimum loop filters for phase-locked loops for different inputs to the phase-locked loop. Their approach is similar to Weiner filter theory, and for a frequency step input, the optimum filter is found to have the form of the active lead-lag filter discussed below.

The first order loop failed with an input response ?( t) = a t, so to provide a matched response to this particular input, we would like a term corresponding to a t. From Equation 2-57, a second order loop requires a loop filter of the form f ( t) = c 0 ?( t) + c 1. The Laplace Transform of this filter is


With the appropriate substitutions, this filter can be rewritten in the form


Three traditional filters for a second order loop are shown in Figure 2.5. Note the active loop filter is identical to Equation 2-59, where we attempted to match the filter's response to the phase input. Any of the filters yields a second order loop, although the active lead-lag filter provides superior performance.


Figure 2.5: Analog Loop Filters [11]

The second order control loop is distinguished by the appearance of a second-degree polynomial in the denominator of Equation 2-48. However, specifying...

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